| L(s) = 1 | + (3.92 + 2.85i)2-s + (−7.68 + 23.6i)3-s + (−2.60 − 8.00i)4-s + (68.6 − 49.8i)5-s + (−97.7 + 71.0i)6-s + (19.5 + 60.2i)7-s + (60.6 − 186. i)8-s + (−304. − 220. i)9-s + 412.·10-s + (−355. + 187. i)11-s + 209.·12-s + (−247. − 179. i)13-s + (−95.0 + 292. i)14-s + (652. + 2.00e3i)15-s + (553. − 401. i)16-s + (478. − 347. i)17-s + ⋯ |
| L(s) = 1 | + (0.694 + 0.504i)2-s + (−0.493 + 1.51i)3-s + (−0.0813 − 0.250i)4-s + (1.22 − 0.892i)5-s + (−1.10 + 0.805i)6-s + (0.151 + 0.464i)7-s + (0.335 − 1.03i)8-s + (−1.25 − 0.909i)9-s + 1.30·10-s + (−0.884 + 0.466i)11-s + 0.419·12-s + (−0.405 − 0.294i)13-s + (−0.129 + 0.398i)14-s + (0.748 + 2.30i)15-s + (0.540 − 0.392i)16-s + (0.401 − 0.291i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.30660 + 0.834796i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30660 + 0.834796i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (355. - 187. i)T \) |
| good | 2 | \( 1 + (-3.92 - 2.85i)T + (9.88 + 30.4i)T^{2} \) |
| 3 | \( 1 + (7.68 - 23.6i)T + (-196. - 142. i)T^{2} \) |
| 5 | \( 1 + (-68.6 + 49.8i)T + (965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-19.5 - 60.2i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 13 | \( 1 + (247. + 179. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-478. + 347. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-44.2 + 136. i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + 1.59e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-2.39e3 - 7.36e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (2.32e3 + 1.69e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (631. + 1.94e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (4.25e3 - 1.31e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.99e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-281. + 865. i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (6.77e3 + 4.92e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (8.95e3 + 2.75e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (8.22e3 - 5.97e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 - 2.54e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (4.08e4 - 2.97e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-5.16e3 - 1.58e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-4.12e4 - 2.99e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-2.84e4 + 2.06e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 - 2.88e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (5.53e4 + 4.01e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.41698541581963872265469718597, −17.97862729611126059954501599682, −16.56554469599906485345162428587, −15.61213058199610227580642755169, −14.35174247455232410910086133635, −12.75103649191053573410416386519, −10.33238058839214768189862570918, −9.391740504381823794693804765511, −5.61216458701510712975259272142, −4.87960287696537153462233331157,
2.31513618489142778535409748856, 5.90137560864310024066212373426, 7.62147200387900039171674046358, 10.67618508830146354976031426523, 12.19103121887286206256811950982, 13.48271327336093740392742141859, 14.04879432999327387756428377608, 17.14751447711182423187263869248, 17.92118925375442229013478449097, 19.07340227815149460078217329817
